Sometimes it is hard for me to decide whether to use "zu" or "um... zu". I think in English whether it should be "to" or "in order to", but it's still often hard to distinguish. For example, consider the following sentences:

(a1) Ich finde jeden Tag zwei Stunden, Deutsch zu lernen.

(a2) Ich finde jeden Tag zwei Stunden, um Deutsch zu lernen.

(b1) Ich verbringe jeden Tag zwei Stunden, Deutsch zu lernen.

(b2) Ich verbringe jeden Tag zwei Stunden, um Deutsch zu lernen.

I don't have any logical way to think about it, but rather just using my sense. I think (a1), (b1), and (b2) are correct. Is it so?

2 Answers
2

Actually, while every sentence is perfectly understandable (though a1, b1 and b2 don't sound correct in my ears, and a2 is incorrect word usage - although grammatically correct), I'd use none of those.

I'd either use:

Ich verbringe jeden Tag zwei Stunden damit, Deutsch zu lernen.

or

Ich finde jeden Tag zwei Stunden Zeit für's Deutsch lernen.

But the shortest way is usually best, so I'd actually use:

Ich lerne jeden Tag zwei Stunden Deutsch.

No need for any zu.

Usually when you use "zu", it's an indication that your sentence is too long, passive voice, or that your formulation is too complicated.

As for the difference between "zu", and "um ... zu":
The word "zu" has several meanings.

It can mean "towards", and as such is some kind of a mixture between preposition and adverb of place, although it technically is a preposition.

"Um zu" can be properly translated with "in order to".
So every time any sentence has some kind of "directional" meaning (e.g. ich gehe zu, zu Gunsten von [die Gunst "geht" zu jemandem] ), it is less likely there is an "um".
This "directional" meaning can also include "quantity" (bis zu, ab und zu) and property (gehören zu, zählen zu).

Every time this "directional" meaning is NOT present, you usually can (or have to) use "um zu".

"zu" can also have the meaning of "too" (as in too many), in which case the the "um" is never used.

Also "zu" can be an adjective (zu as in "Der Laden ist zu." - the shop is closed.), or a part of a noun, in which case "um" is never used.

Use "um" only when

the word is used as a preposition

and

the context does not imply any kind of direction/ownership

or when you'd use "in order to" in English.

Ich brauche das Geld, um nach Hause zu fahren.
I need the money in order to get home.

It's quite hard to answer because it's only a very small nuance that makes a difference (if there is one at all), and colloquially, there is actually no difference to decide for zu or um... zu.

To my mind, um... zu is used if your sentence is like an answer to a Warum, Wofür or Für was question. You'll use it more, if you additionally explain a reason why.

In spite of this, the zu-part of an Infinitiv-mit-zu sentence is closer associated with the first part, like one unit.

Colloquially, I would accept all your 4 sentences, although verbringen almost beats you to use it with damit here. The um alternatives sound a bit more like you want to explain to someone why you've spent the hours, the zu sententences sound a bit more like what you were doing the hours. But as stated in the beginning, you can mean it the other way round as well.